Problem: Solve for $x$ and $y$ using elimination. $\begin{align*}-2x+8y &= -5 \\ -4x-4y &= 5\end{align*}$
Solution: We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $1$ and the bottom equation by $2$ $\begin{align*}-2x+8y &= -5\\ -8x-8y &= 10\end{align*}$ Add the top and bottom equations. $-10x = 5$ Divide both sides by $-10$ and reduce as necessary. $x = -\dfrac{1}{2}$ Substitute $-\dfrac{1}{2}$ for $x$ in the top equation. $-2( -\dfrac{1}{2})+8y = -5$ $1+8y = -5$ $8y = -6$ $y = -\dfrac{3}{4}$ The solution is $\enspace x = -\dfrac{1}{2}, \enspace y = -\dfrac{3}{4}$.